Logic of regulation by two transcription factors¶
Because X and Y both regulate Z in an FFL, we need to specify how they collaborate in the regulation.
For the sake of illustration, let us assume we are discussing C1-FFL, where X activates Z and Y also activates Z. One can imagine a scenario where both X and Y need to be present to turn on Z. For example, they could be binding partners that together serve to recruit polymerase to the promoter. We call this AND logic. In other words, to get expression of Z, we must have X AND Y. Conversely, if either X or Y may each alone activate Z, we have OR logic. That is, to get expression of Z, we must have X OR Y.
So, to fully specify an FFL, we need to also specify the logic, either AND or OR, of how Z is regulated. Including choice of logic gives a total of 16 possible FFLs.
We are now left with the task of figuring out how to mathematically encode AND and OR logic. Before doing so, we note that, as discussed previously, we are using Hill functions, which are phenomenological functions describing how effectors may regulate gene expression capturing both the necessary concentration of effector ($k$) and the ultrasensitivity of the regulation ($n$). When the molecular details of the regulation mechanics of an effector are known, we may derive the appropriate functions describing gene expression regulation rather than using Hill function. Similarly, for two effectors, we could also derive the functions from the molecular details and discover what kind of logic emerges. See, for example, this 2005 paper by Bintu and coworkers. We often do not know the molecular details, and Hill functions and the two-effector variants thereof we present below are quite useful in analyzing the properties of circuit architectures.
We now proceed to formally write mathematical expressions for the dynamics of a gene product Z under regulatory control of effectors X and Y. The dynamics of the concentration of Z may be written as
\begin{align}
\frac{\mathrm{d}z}{\mathrm{d}t} = \beta \,f(x, y) - \gamma z,
\end{align}
where the lowercase letters denote the concentrations of the respective species.
Our goal is to specify the dimensionless regulatory function $f(x, y)$ that encodes how X and Y may together regulate Z. Our approach is to assign a "weight" to each state of a promoter region. With two effectors, X and Y, the promoter region could be unbound, bound with X, bound with Y, or bound with both X and Y. To get the regulatory function, we sum the weights of states that allow polymerase binding and divide by the sum of all weights. This gives the fraction of time that expression of the gene is "on." For example, if X and Y are both activators and they together have AND logic, we have
\begin{align}
f(x, y) = \frac{\text{X and Y bound weight}}{(\text{unbound weight}) + (\text{X bound weight}) + (\text{Y bound weight}) + (\text{X and Y bound weight})}
\end{align}
The weights are chosen to give Hill-like functions.
| promoter region state |
weight |
dimensionless weight |
| unbound |
$$1$$ |
$$1$$ |
|
| X bound |
$$(x/k_x)^{n_x}$$ |
$$x^{n_x}$$ |
|
| Y bound |
$$(y/k_y)^{n_y}$$ |
$$y^{n_y}$$ |
|
| X and Y bound |
$$(x/k_x)^{n_x}\,(y/k_y)^{n_y}$$ |
$$x^{n_x}\,y^{n_y}$$ |
|
The dimensionless weights are given by substituting $x \leftarrow x/k_x$ and $y \leftarrow y/k_y$. We will use the dimensionless versions of these functions henceforth. We note that the denominator of the regulatory function $f(x,y)$ is always the same,
\begin{align}
1 + x^{n_x} + y^{n_y} + x^{n_x} y^{n_y} = (1 + x^{n_x})(1 + y^{n_y}).
\end{align}
Alternatively, we could have a structure where maximally only one of the two effectors may be bound at a time (for example due to steric reasons), in this case the states and weights are given in the table below.
| promoter region state |
weight |
dimensionless weight |
| unbound |
$$1$$ |
$$1$$ |
|
| X bound |
$$(x/k_x)^{n_x}$$ |
$$x^{n_x}$$ |
|
| Y bound |
$$(y/k_y)^{n_y}$$ |
$$y^{n_x}$$ |
|
In this case, the denominator for all of the regulatory functions is $1 + x^{n_x} + y^{n_y}$. We will refer to such regulatory functions as corresponding to "single occupancy."
With this prescription, let us proceed to write the regulatory functions $f(x, y)$ for various architectures.
